Appendix B

1
Appendix B

• Mathematics of Compounding
• and Discounting

2
Compound Interest

• PV present value
• iinterest rate, discount rate, rate of return
• Idollar amount of interest earned
• FV future values
• Other terms
• Compounding
• Discounting

3
Compound Interest

• FVPV (1 i)n
• When using a financial calculator
• n number of periods
• i interest rate
• PV present value or deposit
• PMT payment
• FV future value
• n, i, and PMT must correspond to the same period
• Monthly, quarterly, semi annual or yearly.

4
Compound Interest

• Various Compounding Periods
• Annually FVPV (1 i)n
• Semiannually FVPV (1 i/2)n 2
• Quarterly FVPV (1 i/4)n 4
• Daily FVPV (1 i/365)n 365

5
Solving Problems

• Various Methods
• Formulas
• Financial Calculators
• Tables in Appendix D

6
Future Value of a Lump Sum

• FVPV(1i)n
• This formula demonstrates the principle of
  compounding, or interest on interest if we know
• 1. An initial deposit
• 2. An interest rate
• 3. Time period
• We can compute the values at some specified time
  period.

7
Future Value of a Single Lump Sum

• Example assume Astute investor invests 1,000
  today which pays 10 percent, compounded annually.
  What is the expected future value of that deposit
  in five years?
• Solution 1,610.51

8
Present Value of a Future Sum

• PVFV 1/(1i)n
• The discounting process is the opposite of
  compounding
• The same rules must be applied when discounting
• n, i and PMT must correspond to the same period
• Monthly, quarterly, semi-annually, and annually

9
Present Value of a Single Lump Sum

• Example assume Astute investor has an
  opportunity that provides 1,610.51 at the end of
  five years. If Ms. Investor requires a 10 percent
  annual return, how much can astute pay today for
  this future sum?
• Solution 1,000

10
Annuities

• Ordinary Annuity
• (e.g., mortgage payment)
• Annuity Due in Advance
• (e.g., a monthly rental payment)

11
Future Value of an Annuity

• SR1(1i)n-1 R2(1i)n-2 .. Rn
• Ordinary annuity (end of period)
• Annuity due in Advance (beginning of period)

12
Future Value of an Annuity

• Example assume Astute investor invests 1,000 at
  the end of each year in an investment which pays
  10 percent, compounded annually. What is the
  expected future value of that investment in five
  years?
• Solution 6,105.10

13
Present Value of an Annuity

• PVA R1 1/(1i)1 R2 1/(1i)2...
• Rn 1/(1i)n

14
Payment to Amortize Mortgage Loan

• Same Formula as PV of an Annuity
• PVA R1 1/(1i)1 R2 1/(1i)2...
• Rn 1/(1i)n
• PV is known
• Solve for R
• Amortization Schedule

15
Payment to Amortize Mortgage Loan

• Example assume Astute investor would like a
  mortgage loan of 100,000 at 10 percent annual
  interest, paid monthly, amortized over 30 years.
  What is the required monthly payment of principal
  and interest?
• Solution 877.57

16
Remaining Loan Balance Calculation

• Example determine the remaining balance of a
  mortgage loan of 100,000 at 10 percent annual
  interest, paid monthly, amortized over 30 years
  at the end of year four.
• The balance is the PV of the remaining payments
  discounted at the contract interest rate.
• Solution 97,402.22